We investigate how the performance of search for solving finite constraint satisfaction problems (CSPs) is affected by the level of interchangeability embedded in the problem. First, we describe a generator of random CSPs that allows us to control the level of interchangeability in an instance. Then we study how the varying level of interchangeability affects the performance of search for finding one solution and all solutions to the CSP. We conduct experiments using forward-checking search, extended with static and dynamic ordering heuristics in combination with non-bundling, static, and dynamic bundling strategies. We demonstrate that: (1) While the performance of bundling decreases in general with decreasing interchangeability, this effect is muted when finding a first solution. (2) Dynamic ordering strategies are significantly more resistant to this degradation than static ordering. (3) Dynamic bundling strategies perform overall significantly better than static bundling strategies. Even when finding one solution, the size of the bundles yielded by dynamic bundling is large and less sensitive to the level of interchangeability. (4) The combination of dynamic ordering heuristics with dynamic bundling is advantageous. We conclude that this combination, in addition to yielding the best results, is the least sensitive to the level of interchangeability, and thus, indeed is superior to other searches.